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Applied Quantitative Methods

Applied Quantitative Methods. Lecture 5 . Statistical Inferences. October 20 th , 2010. Last Lecture Review. Statistical inferences. Estimator (Sample statistic): a function of sample variables TE: sample mean, sample standard deviation Estimate – a realized value of an estimator

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Applied Quantitative Methods

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  1. Applied Quantitative Methods Lecture 5. Statistical Inferences October 20th, 2010

  2. Last Lecture Review

  3. Statistical inferences • Estimator (Sample statistic): a function of sample variables • TE: sample mean, sample standard deviation • Estimate – a realized value of an estimator • Sampling distribution: a likelihood of various outcomes of the estimator across different random samples • Statistical inferences • Estimation: providing a range of potential values (confidence interval) for a population parameter based on a sample statistic • - Point & Interval estimators • 2)Testing: using a sample to test the claim about the value of a population parameter

  4. Point Estimation • Method of moments • Equating first k sample moments to the first k population moments and solving the resulting system of equations • X1…..Xn ~iid from a population with f(x|θ1…..θk) • TE Let X1…..Xn ~iid N(θ,σ2)

  5. Point Estimation(Cont.) • Maximum likelihood • For each sample point X, let be a parameter value at which attaints its maximum as a function of θ. • Likelihood function: • - is a random variable • TE: Let X1…..Xn ~iid N(θ,1)

  6. Properties of Estimators • Finite sample properties • Unbiased estimator • The bias is systematic Source: Doane & Seward (2009) Applied Statistics.

  7. Properties of Estimators (Cont.) • Sample mean is an unbiased estimator of the population mean • Sample variance is an unbiased estimator for the population variance • TE: Sample mean vs. First observation in the sample • and • => A need for additional criteria

  8. Properties of Estimators (Cont.) • 2. Efficiency: Unbiasedness + Minimum variance • - Sampling distribution of an estimator should have a small standard error • Sampling variance: • TE: For normal distribution, sample mean and sample variance are MVUEs.

  9. Properties of Estimators (Cont.) • Asymptotic properties of estimators • Consistency: is a consistent estimator of θ if for every ε > 0 • If is consistent, then • A minimal requirement • An unbiased estimator • which • is consistent

  10. Properties of Estimators (Cont.) • 2. Asymptotic normality: shape of the sampling distribution • Central Limit Theorem • Let {X1…..Xn} be a random sample from a population with mean μ and variance σ2, then • The rule of thumb: n ≥ 30 is sufficient to assume normality for a symmetric or slightly skewed population without outliers

  11. Properties of Estimators (Cont.)

  12. Interval Estimation • Confidence interval • Centered on a point estimate • The size of the confidence interval is determined by: • Confidence level (100(1-α) %): more confidence – less precision • Size of σ: larger σ – wider interval • Sample size n: larger n – narrower interval • Methodology of estimation • Case 1: known σ • Case 2: unknown σ

  13. Interval Estimation (Cont.) • Confidence interval for sample mean • CLT: Sampling distribution of follows normal distribution with mean μ and as n increases. • Case 1: known σ • TE: Weekly hours of TV watching • Sample n =25, and a known σ = 10 • Find a 95% confidence interval for the population mean μ • Margin of error: • Critical values (z-scores): • Confidence interval:

  14. Interval Estimation (Cont.) • Case 2: known σ • TE: n=25, and s = 15 • Find a 95% confidence interval for the population mean μ • Margin of error: • Critical values (t-scores): • Confidence interval: • N!B! Interpreting confidence intervals • The probability that μ is in the interval is 95%

  15. Interval Estimation (Cont.) • A 95% confidence interval for the population mean is • - In 95 % of random samples, confidence interval would contain μ

  16. Hypotheses Testing (HT) • Hypothesis is a statement about the value of a population parameter • Hypothesis testing (HT): consistency of data with an assumption about population parameters • - One-sample & Two-sample HT • 1) One-sample HT • The null hypothesis (H0): claim about the value of a population parameter • The alternative hypothesis (H1): • Nondirectional (two-sided): • Directional (one-sided):

  17. Hypotheses Testing (Cont.) • Step-by-step procedure • Formulate H0 and H1 • - mutually exclusive statements • H0 always refers to the population • H1 always refers to a sample • Assume that H0 is true • A benchmark value (prior knowledge, standards)

  18. Hypotheses Testing (Cont.) • 3. Level of significance (α) acceptable level of risk • - α is a probability of Type I error • Conventional levels of significance: 0.01, 0.05 and 0.1 • Confidence level: (1- α)*100% • Type II error (β): • α and β are inversely proportional • To decrease the likelihood of both errors we need to increase the sample size

  19. Hypotheses Testing (Cont.) • 4. Critical values: region of rejection for H0 • For known σ: z-score (standard normal distribution) • For unknown σ: t-distribution • Critical values depend on: • type of the test (one-tailed or two-tailed) • and level of significance (α) • 5. Collect sample • - calculating sample statistics

  20. Hypotheses Testing (Cont.) • 6. Test statistic • Case 1: known σ • Z-score: • Case 2: unknown σ • t-score: • Test statistics measure how far is the value of sample statistic from H0

  21. Hypotheses Testing (Cont.) • 7. Drawing conclusion • Rejection region: the area under the sampling distribution curve that defines an extreme outcome • Left-tailed test: H0 is rejected if zcalc< - zα • Right-tailed test: H0 is rejected if zcalc > zα • Two-tailed test: H0 is rejected if zcalc< - zα/2 or zcalc > zα/2

  22. Hypotheses Testing (Cont.) • One-sample hypothesis test • TE Quality control • Industry standard 216 mm. • Random sample n=50, =216.007, σ=0.023, 5% level of significance • State hypotheses • H0: μ0 = 216 • H1: μ0 > 216 • Level of significance α=0.05 • Critical values of z statistic: • Right-tailed test: z0.95=1.645 • Sample mean 216.007 • Test statistic: • 6. Decision: zcalc falls into rejection region. H0 is rejected at 5 % level of significance

  23. P-value Approach • P-value: What is a probability that we would observe a particular sample mean (or a value even father away from μ0 ), if Ho is true? • - is compared to the level of significance α • Decision rule (right-tailed test): • Reject H0 if P(Z > zcalc)< α, otherwise fail to reject H0 • - the smaller the p-value, the stronger the evidence for rejection H0 • TE zcalc=2.152 • p-value = P(Z > 2. 152) = 1-P(Z< 2.152) = 1-0.9843 = 0.0157 • 0.0157 < 0.05 => H0 is rejected • Interpretation: in a right-tailed test a test statistic of zcalc=2.152 (or a more extreme value) would happen by chance about 1.57 percent of time if H0 is true • Two-tailed test: Reject H0 if 2 P(Z > zcalc)< α, otherwise fail to reject H0

  24. Hypotheses Testing • 2) Two-sample HT • TE Research question: does attendance of lectures boost learning • H0 : There is no difference in midterm test average scores of students who attended lectures and who did not • H0: μ1= μ2 • H1(nondirectional): The average test score in two groups of students will be different • H1:μ1≠ μ2 • Test statistics for three cases: • - both population standard deviations are known • - both population standard deviations are unknown • - both population standard deviations are unknown but equal • Z-score:

  25. Hypotheses Testing Case 1: Margin of error for known σ1 and σ2 : Case 2: Margin of error for unknown σ1 and σ2 :

  26. Next Lecture N!B! Midterm exam: October 27th, at 9:15, RB 212 1. Waldman, M., Nicholson, S. & Adilov, N. (2006). Does Television Cause Autism? NBER Working Paper 12632. 2. Mankiw, Romer and Weil (1992). A Contribution to the Empirics of Economic Growth. The Quarterly Journal of Economics, pp. 407.

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